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Problem 31 Hard Difficulty

Use a graphing calculator or computer to graph both the curve and its curvature function $\kappa(x)$ on the same screen. Is the graph of $\kappa$ what you would expect?
$$y=x^{-2}$$

Answer

$$
\kappa(x)=\frac{\left|6 x^{-4}\right|}{\left[1+4 x^{-6}\right]^{\frac{3}{2}}}
$$

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Video Transcript

all right. In this problem, we went to look at the graph of the function. Why equals X to the power of negative two and its curvature on the same graphing window. So that way we can compare the two. So to find the curvature, which we need to do first, I'm going to use the formula in the box on the right hand side. So we're gonna need the first and second derivatives. Why? So why prime is going to be be equal to negative two times X to the power of negative three and why double prime is going to be equal to six times X to the power of negative four. And then if I use my formula for capital of X in the numerator, we're gonna have the absolute value of six times X to the power of negative four over one plus negative two X to the power of negative three, which is our first derivative that's gonna be squared. And then the entire denominator is gonna be raised to the power of three halves. I'm not really going to simplify this because we're just gonna be plugging it into the calculator. So the point of this problem is not to know how toe simple fight with algebra, it's toe. You know how to enter it in the calculator and compare the two functions. So first thing we'll do is put in the graph of why into our calculator. So ah, here for why one and I get to this by going to the y equals menu on my t 84. For why one we're gonna have X to the power of negative, too. That's gonna be our graph of why I'm gonna just my window a little bit have already pleaded with this. Um but you might have to do the same, but I found that this is best viewed on the domain from negative 4 to 4 for X coordinates, all one tick mark apart and from negative to Teoh. Positive eight on our Y coordinates for a range. So click grafts. We can see what this looks like, and we see a function that has a vertical assam toot at X equals zero su. What we want to do is we want to see if we can figure out an idea of what our curvature function might look like before we actually graph it who remember that curvature is how quickly why is changing directions. So what I see is that our function looks If we're looking at it, coming from X equals negative infinity going this direction, we see that we started off with kind of a flat curve. And then somewhere between X equals negative two and X equals negative one. Our function increases sharply, so it changes directions. And then once it kind of gets closer to that vertical Assam to it, it stays pretty flat again. But now it's going vertical. So we expect to have Ah, Matt, a relative maximum for curvature somewhere between the point X equals negative one and X equals negative too. So well, you'd say, is that why changes direction most quickly between? And we said that's between X equals negative to and X equals negative one, and we see that reflection on the other side of the graph. So coming from our vertical ascent to zero, the closer that X is to that vertical assam toe. It's not really changing directions, but once they get somewhere between X equals one and X equals toots, it really starts toe to turn a bit, so it also changes most quickly between X equals one and X equals two. So what we would expect to see as for capital of X T have to relative maximum because our function changes directions the most at these two different points. So let's go ahead and graft Kappa of X so we can see what it looks like. I'm gonna go back to my y equals menu. I'm gonna go down to function. Why, too, scroll over to the left and changed my line style by clicking Enter on that flashing bar. So now we have a bold style. It's gonna make it a little easier. Look at we in the numerator going have absolute values after goodbye mass menu here to the numbers, uh, sub menu and click enter for my absolute value, we're gonna have six times X to the power negative four in the numerator that's divided by one plus negative two x to the power of negative three. That quantity is going to be squared and then the entire denominator is being raised to the three house power. And if I click graph, we're going to see our graph of capital Vex show up here and we do see that we have too little bumps and the graphs it to relative maximum. I'm actually going to zoom in a little bit. So if I get in my y equal or my window menu and I'm gonna change my why max value to just so we can zoom in a little bit, see what's going on there. If I click graph, we see that the bold function, which is capital of X of the curvature, does have a peak somewhere between X equals negative to an X equals negative one and again between one and two. X equals wanted to. So this does look like the graph of cap of X of the graph of the curvature that we would expect to see for this assumption.

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